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Line of Sight and Wargames

In my designer’s blog, I wrote a series of entries about line of sight in wargames. If it came out well, it was always my intent to edit and assemble it into an essay for the design section of the sight. As it happened, I thought it came out pretty well, and so here it is.

Introduction

Line of sight is one of the classically hard problems to solve in wargame design. This essay looks at line of sight to see what makes it so hard, and to suggest a new approach that might prove more fruitful than the traditional wargame approach to the problem.

The Traditional Approach

To begin the discussion, let’s review how wargames normally handle line of sight:

The above example is a trivial wargame line of sight problem: There is no blocking terrain between the cannon and the target, and they are both on the same elevation, so the line of sight is clear. The cannon can see and fire at the target. Let's take a look at the situation in profile, so we can see what is being modeled:

Now, far be it from me to say that there is nowhere in the world that isn’t exactly like this, but much more common are cases where the ground has gentle undulations, standing fields of crops, tall grass, patches of trees and bushes, fences, and so on. Wargames in general don’t show any vegetation less significant than actual forests, and ignore any hill less than 10 meters high. Now, the problem here it only takes 2 meters in height of stuff to hide a human target, and the longer the distance you try to cover, the more likely you’ll find something 2 meters or more in height that does exactly that. And so, what appears to be an easy line of sight problem is only easy because the game leaves out all the things that would make it complicated — and which could give the opposite answer as to whether or not the cannon could see and fire at the target:

Part of the problem here is of course that wargame maps just plain don’t have the detail required to model “flat” ground well, and not just a little short: really, they would need to increase the vertical map detail by about 10x in order to have the raw map data needed. Where things get sticky is that that level of map detail doesn’t usually exist in the sources themselves. Where things get stickier is that the rules modeling to take advantage of more detail would greatly complicate and slow down play.

The above example, only deals with the problem of “flat” ground. What about ground that isn’t flat?

Before beginning, let’s get a trivial matter out of the way. Absolute elevation (as in height above sea level) is of no importance whatsoever in line of sight. The two examples below are exactly the same, even though they occur at different elevations:


Now, I don’t know of any wargames that have trouble with this. Everybody knows that absolute elevation doesn’t matter, and everybody gets it right without any difficulty. Now, let’s take a look at relative elevation, when the ground isn’t flat and the cannon and target are at different heights. Here is how that might look in a wargame that includes elevation contours:

And now let’s take a look at what the above example represents in cross-section:

The interesting thing is that the above cross-section is just our old friend, the flat ground cross-section, just tilted. Relative elevation, as such, made no difference either to line of sight. Of course, the ground would be unlikely to be really this straight a line, no more than it would in the flat ground example. Really, it would likely be more like this:

Which is just our old friend, the real-world “flat” ground, just tilted. So, as it turns out, not only do we not care about the absolute elevation, we don’t care about relative elevation either — at least, we don’t care about relative elevation when we are talking about just the firer and the target. We only get useful line of sight data when we include the elevation of a third point: a potential obstacle.

For the moment, let’s forget about the ground clutter that is below our game’s map resolution and just take a look at the big stuff, such as a hill tall enough to actually show on a wargame map. For our first example, let’s go back to having the firer and target on the same elevation, and introduce a big hill between them. We’ll see how it looks on the wargame map and what it represents (ignoring real-world ground clutter):


Now, so far, so good, right? We can easily see on the wargame map that with the cannon and the target at 590 yds. elevation, and a hill at 600 yds. elevation between them, that there is no line of sight.

But what about a more complex problem? What about the very common case where the cannon and the target are not at the same elevation, what then?

The problem of line of sight where the shooter and the target are not on the same elevation has been where wargame line of sight tends to simply break down. The mathematics of calculating whether a given obstacle blocks the line of sight to a given target can be expressed in a variety of different ways, but basically amounts to comparing the vertical angle from the shooter to the target and comparing it to the vertical angle from the shooter to the obstacle to see which is greater.

Comparing the vertical angles requires comparing ratios of the difference in elevation between the shooter and the target, the shooter and the potential obstacle, the horizontal distance between the shooter and the target, and the shooter and the potential obstacle. Since the calculation will require multiple decimal places of accuracy, even though it is simple in theory, it is far to computationally complex for a boardgame.

Worse still, however, is the fact that to give the right answers it requires the correct elevations; the approximate elevations used in board games give rise to a variety of odd effects. For example, here is the illustration we used earlier for a smooth slope:

Plugging its numbers literally into the line of sight formula named above represents the smooth slope not as smooth, but as a sort of ziggurat:

Each “step” in the ziggurat creates a blind spot near the step where the corner of the step obscures line of sight in both directions, blind spots that only exist because approximate elevations were used. To compare the vertical angles, we can’t get decent answers with approximate elevations. We would need to add interpolations for the heights between the elevation steps and adjusters for the height of the shooter and targets themselves. The whole exercise is just impossible and collapses under its own weight.

To review: There are two problems with line of sight in boardgames. The first is that they have too little data to give good answers on “flat” ground, and can’t use the data they do have on sloped ground because of overwhelming computational complexity. Faced with this problem, one can understand why the designer, in the line of sight rules for the third edition of Three Days at Gettysburg, wrote: “We’ll try to keep this as simple and basic as possible...There are sure to be anomalies: try to solve them based on the underlying principles these rules portray.” Now this is about as clear an example as you’re likely to find of a designer simply throwing up his hands in despair (and this is after three editions of the game and two editions of its predecessor, Terrible Swift Sword) and leaving it up to the players to fix the game.

Now, I wouldn’t want you to think that by mentioning Three Days at Gettysburg above I was singling it out for abuse. Actually, I think the rules text just acknowledged that the line of sight methodology it (and many, many other games as well) used had come to a sort of dead-end: not really satisfactory, but with no obvious way forward. I don’t think that TDAG had worse line of sight rules than other games; it was just more frank about its limitations.

So, what is to be done? Can anything be done? Obviously I think so. The whole premise of this essay is that something can be done.

Starting Over

Now, we have seen that the way we have been going has led to a dead end. So, let’s go back to the beginning and take a fresh look at the problem.

(1) If the ground were perfectly flat, then line of sight would have no definite limit. Of course, minimally, the earth is round, not flat (a point that seldom matters on land but which matters very much at sea). The reason that the earth's shape limits line of sight is that it is a convex curve. Now, what is interesting is that any convex curve will do this, not just the convex curve produced by the shape of the earth. This is because the curve of the ground rises up and blocks the line of sight from the shooter to the target.

(2) In previous entries, we talked about ground clutter: small (less than 4 meter) variations in elevation, standing fields of crops, tall grass, patches of trees and bushes, fences, etc. Ground clutter is limits line of sight, even on “flat” ground. Now, what enables a shooter to see over ground clutter is when the shape of the earth is concave, dropping the clutter below the line of sight from the shooter to the target.

(3) So, to build on this principle, rather than try and identify ground first by elevation, let’s start by identifing it by general ground shape. Areas of ground will be identified as either convex or concave: hill or valley.

The Guns of Gettysburg

In fact, this is the first-order basis for the map of The Guns of Gettysburg. The ridges and hills on the battlefield are positions while the valleys are the areas. This version of the GoG map, with non-ridge positions stripped out (we’ll talk about non-ridge positions in a later blog entry) shows you how this theory was applied to create the map design of an actual game. (As a side note, I also left in obstructed posititions, since they have their own large role to play in line of sight.)

Gettysburg Hills

(Click on the image to go to open it in its own window.)

The importance of the ridges on the above map to the battle of Gettysburg can be seen by this map showing on it the main defensive lines the Union army took during the course of the battle:

Essentially, the battle of Gettysburg was a series of fights for ridge-lines. If the Union lost one, they fell back to another. Further, Confederate attacks typically consisted of setting up on the ridge line opposite the Union, and then attacking across the valley in between their ridge line and the Union ridge line. The critical nature of the ridges to the battle of Gettysburg is almost impossible to overstate. So, in the previous entry we proposed a map system which differentiated terrain first by whether it was convex or concave and built the Gettysburg map on that principle. Was it a success?

Not entirely.

There are several problems with the map as shown above. First, it has no mechanisms for regulating movement: it cannot answer the simple question of how far a unit can move. Second, the map cannot represent the important tactic of flanking a ridge position, forcing the enemy to defend across floor of a valley in order to protect his flank and rear. Third, the map has no mechanism for regulating fields of fire, both in terms of range and firing angle. All of these things, taken together, required non-ridge positions, which were added to make the final map:

Still, as a first order solution to the problem of line of sight, simply differentiating between convex and concave ground proved remarkably successful. (I think.) There is one thing, however, that the map design did not naturally handle well, and that is the case where fire went from one ridge to another, over an intervening lower ridge. Now, because of the limits of weapon ranges, this was much less of a problem than it might seem. Really, there are only a few places on the map where this is possible, and those were handled by the dotted special line-of-sight lines on the map.

In any case, the result is a map which can handle complex line of sight in a math-free way. To a considerable extent, that is because line-of-sight calculations are built into the map design itself. The map does the work, so that the players don’t have to.

Better Line of Sight on a Hex Map

Now, the map for GoG was specially designed to accomodate line of sight. But what about normal hex grid maps? Could they use the same system? Further, is GoG the best that can be done, or is it possible to do even better? The answers, I think, are yes, no, and yes.

As with any game, GoG is a compromise among multiple competing goals. Accurate line of sight was a goal, but not the only goal, and consequently the game’s line of sight rules were limited in some ways in order to achieve other desired features of the game, such as design targets for scale, complexity, and playing time.

The principles underlying the line of sight rules, however, do not depend on any of the distinctive design features of GoG. We could just as easily apply them to a hex-grid based game at different scale. By way of example, below you can see a portion of the GoG map with a hex grid overlaid onto it. This grid is close to the map scale of Three Days at Gettysburg, though I haven’t attempted to make it an exact match.

Now, a hex grid of this scale would require a much larger map than GoG possesses, or REALLY small pieces to fit in such small hexes, but it does show how a different scale and terrain model can result in higher resolving power. For example, the spur at the northern end of Cemetery Hill can be resolved by the hex grid as a half-dozen or so hexes, but cannot be resolved at all by the areas of GoG due to the areas’ larger size. (Of course, hexes have their own issues, but our purpose here is to demonstrate how to apply the LOS system to a different style of game map, not to get into the various pluses and minuses of hex grid vs. area designs. That is a discussion I may take up at some future date, but not now.)

Anyway, before we start, I want to try to break the spell of the arbitrary contour intervals in the map. Because we can see 20-foot intervals on this map and we can’t see any others, we find it very hard to resist the idea that the location of these intervals is important, even though we know, that if the mapmaker had put in intervals at 630, 610, 590, 570 yards, etc. instead of 620, 600, 580, and 560, we would be looking at lines where we now see gaps and gaps where we now see lines. One possible antidote is this map of the same area, with many more contours, and none of them privileged with special markings:

One of the things we can see on the above map is that there is actually a good-sized hill on Cemetery Ridge, south of Cemetery Hill proper, which is completely invisible on the map with contours only at 20-foot intervals. What appears on the 20-foot contour map as a large plateau south of Cemetery Hill is actually an illusion created by lack of detail: The “plateau” actually consists of a hill and the meeting of two ridge lines, with the beginning of a valley between them that deepens as it goes north.

The importance of not being hypnotized by the location of contour lines is because to build our LOS map, what we want to find are the ridges and valleys: the convex ridges that will block fire across them, and the concave valleys where ridge-to-ridge fire over them is unimpeded by ground clutter. So, let’s make a map that marks convex vs. concave rather than specific elevations:

Now, this is a rough-sketch, and if I were actually designing the game rather than writing an essay as a design exercise, I would wat to review this map carefully. As it is, however, it will do to make my point. Simple as this is, it catches a large percentage of the important line-of-sight features of this part of the battlefield.

Before improving it, let’s consider what the basic rules should be. First, we need to take into account ground clutter. Since we don’t have the map accuracy to depict it, in our back-of-the-envelope game design, we will plan for reduced fire range and effectiveness by units firing from valley hex to valley hex (as well as a lesser penalty for firing from valley hex to ridge hex). Second, we will not permit fire (or at least aimed fire: our game might or might not permit unaimed fire) from a valley over a ridge into another valley (a valley-ridge-valley sequence), or aimed fire in ridge-valley-ridge-valley sequence (either direction).

With these rules in mind, we can see one deficiency already: Cemetery Hill’s line of sight is obstructed by the low ridge to its northwest, but on the actual battlefield Cemetery Hill is high enough to see over it. Now, one way to fix this is by basically the same method we used on the GoG area map: we mark off areas (for valleys and hills) on top of the hex grid, and then use special line of sight lines to indicate exceptions. And really, this would work fine. However, because this is a design exercise, and we’ve already seen that method, let’s use a different method: we’ll mark off some ridge hexes as being “dominant hills” that can see over other ridges and down into the valleys behind them, like so:

Now frankly, this method is less flexible than the exception method, but it takes less map markup and is probably easier for players to see. If you don’t need the additional flexibility, this would be the way to go. Now, one thing worth noting about this: we are not indicating absolute elevation, but local prominence. The Gettysburg battlefield, for example, generally slopes upward from the southeast to the northwest corner. An absolute elevation that is high ground on the southeast corner of the map is low ground on the northwest corner. As we only care about local dominance, it is perfectly possible that the same elevation could be dominant in one part of the map and not dominant in another, depending on the height of the surrounding terrain.

Now this map is quite a bit better, but at this scale we should really be thinking about not just capturing the tops of the ridges, but capturing the forward vs. the reverse slope. Military units do not generally deploy on the very tops of ridges; they deploy slightly lower, on what is called the “military crest”, where they are not silhouetted against the sky and the view of the ground at the base of the ridge is better. The following illustration shows this:

Now the interesting thing about the military crest for our purposes is there are two of them, one on each side of the ridge. Depending on the game scale, the difference between the topographic crest and the two military crests may or may not be worth capturing. At this scale, we can reasonably represent it, so let’s do so. We’ll mark the ridge tops with a double line of hexes, for the forward and reverse slope military crests, with a line between them marking the topographic crest. (We don’t need to use a hex to mark the topographic crest, since it is wrong for the scale, and units don’t occupy it anyway.) We will also amend our rules to block line of sight across a topographic crest except from an adjacent hex or from a non-adjacent dominant hill.

Now, the above is a pretty good LOS map of the area, with simple rules and very good results. The map shows ridge positions, forward and reverse slopes, and dominant positions like Cemetery Hill. Further, with the ground-clutter fire penalties for valleys, the rules will strongly incline players towards the sort of tactics that the armies historically adopted. If even better results were desired, they could be obtained for only a modest effort. The use of areas and line of sight exceptions used by GoG, for example, are a very flexible tool that this design doesn’t even use, but could use if desired.

Conclusion

Now, I think I have demonstrated a better way to treat line of sight than is generally used in wargames, but I do not suppose that this is the best that can be done. In discussing this approach, it is my wish to do three things: (1) To give designers an alternative to the traditional approach, (2) to encourage them to see if they can take this basic approach and push it even farther than I have done, either in terms of making it more accurate or easier to use, or both, and (3) to encourage them to think that if one new approach can do better, that with some imagination perhaps they can come up with an entirely new approach of their own that is even better than the approach described above.